Upcoming Seminars & Events
Please note different day and location (Monday, Fine 110.)
In this talk I will discuss both local and global properties of the Allen-Cahn equation in closed manifolds.
We describe a computational framework for computing hybrid traveling-standing waves that return to a spatial translation of their initial conditions at a later time. We introduce two parameters to describe these waves, and explore bifurcations from pure traveling or pure standing waves to these more general solutions of the free-surface Euler equations. Next, we combine Floquet theory in time and Bloch theory in space to study the stability of traveling-standing waves to harmonic and subharmonic perturbations. For the latter, we have developed new boundary integral methods for the spatially quasi-periodic Dirichlet-Neumann operator. While much is known about the spectral stability of pure traveling waves, this is the first study of general subharmonic perturbations of pure standing waves. Our unified approach for traveling-standing waves simplifies the eigenvalue problem that arises in the pure traveling case as well. We conclude with a discussion of general quasi-periodic solutions of the free-surface Euler equations and present preliminary calculations of some simple cases.
The IAS recently organized an Emerging Topics Working Group on " Applications to modularity of recent progress on the cohomology of Shimura varieties". Based on ideas of Calegari and Geraghty, and using recent results of Khare-Thorne and Caraiani-Scholze, the group (Allen, F.Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Thorne and myself) were able to prove modularity lifting theorems for n-dimensional Galois representations over CM (or totally real) fields without a (conjugate) self-duality hypothesis. Most notably our theorems apply in the `non-minimal' case. As applications we prove that elliptic curves over CM fields become modular after a finite base change. We also prove that cohomological (for trivial coefficients) automorphic forms on GL(2) over a CM field satisfy the Ramanujan conjecture. We are not able to reduce the Ramanujan conjecture to the Weil conjectures, rather we deduce it from the potentially automorphy of symmetric powers.
Cartesian products have embedded within them certain natural subsets which are indexed by combinatorial information. The geometric characterization of these subspaces, known now as polyhedral products, has application in toric geometry and topology, combinatorics, geometric group theory, number theory, homotopy theory and arachnid mechanisms. The workshop will focus on the topology and geometry of these and related spaces.